Note that the difference between
\(\frac{1}{N}\sum _{(x,y) \in S} \ell _{\mathrm {log}}(y,f(x)+\Delta _{y})\) 
and
\(\sum _{y\in \lbrace -1,1\rbrace }\frac{\mathbb {P}(y)}{|S_{y}|}\sum _{x\in S_{y}}\ell _{\mathrm {log}}(y,f(x)+\Delta _{y})\) 
vanishes as \(N \rightarrow \infty \) .
The former thus approximates the empirical error
of the logit-adjusted loss \(\ell _{\rm mar}\) , and aligns with our formulation in (REF ).
Further note that while the generalisation error on the left-hand side
of the bound is with respect to the joint distribution (which depends on the skewed label distribution \(\mathbb {P}(y)\) ), it is consistent for minimising the balanced error where the label distribution is uniform when \(\Delta _y = \log \frac{\mathbb {P}(y=+1)}{\mathbb {P}(y=-1)}\)  [1]}.
